What is the value of $\dfrac{d}{dx}\csc(x)$ at $x=\dfrac{\pi}{6}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2$ (Choice B) B $\dfrac{1}{2}$ (Choice C) C $-\sqrt{3}$ (Choice D) D $-2\sqrt{3}$
Solution: Let's first find $\dfrac{d}{dx}\csc(x)$. Then, we can evaluate it at $x=\dfrac{\pi}{6}$. Recall that the derivative of $\csc(x)$ is $-\dfrac{\cos(x)}{\sin^2(x)}$, or $-\csc(x)\cot(x)$. Put another way, $\dfrac{d}{dx}[\csc(x)]=-\dfrac{\cos(x)}{\sin^2(x)}=-\csc(x)\cot(x)$. [Is there a way to know this without memorizing?] Now let's plug in $x={\dfrac{\pi}{6}}$ : $\begin{aligned} &\phantom{=}-\dfrac{\cos\left({\dfrac{\pi}{6}}\right)}{\sin^2\left({\dfrac{\pi}{6}}\right)} \\\\ &=-\dfrac{\dfrac{\sqrt{3}}{2}}{\left(\dfrac12\right)^2} \\\\ &=-{\dfrac{\sqrt{3}}{2}}\cdot {\dfrac41} \\\\ &=-2\sqrt{3} \end{aligned}$ In conclusion, the value of $\dfrac{d}{dx}\csc(x)$ at $x=\dfrac{\pi}{6}$ is $-2\sqrt{3}$.